As an operator on \(L^{1}({\Sigma })\)  and \(L^{\infty }(\Sigma )\) ,
\(E^{\mathcal {A}}\)  is idempotent and
\(E^{\mathcal {A}}(L^{\infty }(\Sigma ))=L^{\infty }(\mathcal {A})\)  and
\(E^{\mathcal {A}}(L^1(\Sigma ))=L^1(\mathcal {A})\) . Thus it can be
defined on all interpolation spaces of \(L^1\)  and \(L^{\infty }\)  such
as, Orlicz spaces [1]}. If there is no possibility of
confusion, we write \(E(f)\)  in place of \(E^{\mathcal {A}}(f)\) . We list here some
of its useful properties:
